Germany TST + pre-TST 2002-20 (V-AIMO) 38p

geometry problems from Germany IMO Team Selection Tests (TST)
with aops links in the names 

(only those not in IMO Shortlist)
[Aimo stands for TST, Vaimo for pre-TST]

2002 - 2019

2002 German TST p2 (VAIMO)
Prove: If $x, y, z$ are the lengths of the angle bisectors of a triangle with perimeter 6, than we have:
$\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} \geq 1.$

2003 German TST p2 (VAIMO)
Given a triangle $ABC$ and a point $M$ such that the lines $MA,MB,MC$ intersect the lines $BC,CA,AB$ in this order in points $D,E$ and $F,$ respectively. Prove that there are numbers $\epsilon_1, \epsilon_2, \epsilon_3 \in \{-1, 1\}$ such that: $\epsilon_1 \cdot \frac{MD}{AD} + \epsilon_2 \cdot \frac{ME}{BE} + \epsilon_3 \cdot \frac{MF}{CF} = 1.$

2004 German TST 1 p2 (VAIMO)
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$. Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.

2004 German TST p1 (AIMO 3)
Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

2004 German TST5 p1 (AIMO)
The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

2004 German TST6 p2 (AIMO)

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that $\measuredangle MPA = \measuredangle BPN    \quad \text{and} \quad    PA \leq PB.

(i. e. give an explicit description of these points without using the points $M$ and $N$).

2004 German TST7 p2 (AIMO) (by Arthur Engel; part of AMM problem #10874)

Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram. 

2005 German TST p9 (VAIMO)
Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that
(a) $\triangle ABC$ is acute.
(b) $a+b+c > r+r_a+r_b+r_c$.

2005 German TST4 p2 (AIMO) (by Arend Bayer)
Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

2005 German TST5 p3 (AIMO)
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.

2005 German TST6 p3 (AIMO)
Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality
$9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,$ and determine when equality holds.

2006 German TST p1 (VAIMO)
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

2006 German TST4 p2 (AIMO) (ISL 1986 USS)
Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.

2006 German TST6 p3 (AIMO)
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at a point $X$. The circumcircles of triangles $ABX$ and $CDX$ meet at a point $Y$ (apart from $X$). Let $O$ be the center of the circumcircle of the quadrilateral $ABCD$. Assume that the points $O$, $X$, $Y$ are all distinct. Show that $OY$ is perpendicular to $XY$.

2006 German TST7 p1 (AIMO)
Let $ABC$ be an equilateral triangle, and $P,Q,R$ three points in its interior satisfying $\measuredangle PCA = \measuredangle CAR = 15^{\circ},\ \measuredangle RBC = \measuredangle BCQ = 20^{\circ},\  \measuredangle QAB = \measuredangle ABP = 25^{\circ}.$ Compute the angles of triangle $PQR$.

2006 German TST7 p2 (AIMO)
The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.

2007 German TST p3 (VAIMO)
A point $P$ in the interior of triangle $ABC$ satisfies
$\angle BPC - \angle BAC = \angle CPA - \angle CBA= \angle APB - \angle ACB.$
Prove that $\bar{PA} \cdot \bar{BC}= \bar{PB} \cdot \bar{AC} = \bar{PC} \cdot \bar{AB}.$

2007 German TST5 p3 (AIMO)
Let $ABC$ be a triangle and $P$ an arbitrary point in the plane. Let $\alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $F.$ Prove:
${AP}^2 \cdot \sin 2\alpha + {BP}^2 \cdot \sin 2\beta + {CP}^2 \cdot \sin 2\gamma \ge 2F$ When does equality occur?

2007 German TST6 p3 (AIMO)
In triangle $ABC$ we have $a \geq b$ and $a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $s_a$ to the altitude $a_a.$ When do we have equality?

2008 German TST p2 (VAIMO)
Let $ABCD$ be an isosceles trapezium with $AB \parallel  CD$ and $\bar{BC} = \bar{AD}.$ The parallel to $AD$ through $B$ meets the perpendicular to $AD$ through $D$ in point $X.$ The line through $A$ drawn which is parallel to $BD$ meets the perpendicular to $BD$ through $D$ in point $Y.$ Prove that points $C,X,D$ and $Y$ lie on a common circle.

2008 German TST4 p3 (AIMO) (by Christian Reiher)
Let $ABCD$ be an isosceles trapezium. Determine the geometric location of all points $P$ such that $|PA| \cdot |PC| = |PB| \cdot |PD|.$

2008 German TST5 p2 (AIMO) (by Gunther Vogel)
For three points $X,Y,Z$ let $R_{XYZ}$ be the circumcircle radius of the triangle $XYZ.$ If $ABC$ is a triangle with incircle centre $I$ then we have: $\frac{1}{R_{ABI}} + \frac{1}{R_{BCI}} + \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} + \frac{1}{\bar{BI}}+\frac{1}{\bar{CI}}.$

2008 German TST6 p1 (AIMO)
Let $ABC$ be an acute triangle, and $M_a$, $M_b$, $M_c$ be the midpoints of the sides $a$, $b$, $c$. The perpendicular bisectors of $a$, $b$, $c$ (passing through $M_a$, $M_b$, $M_c$) intersect the boundary of the triangle again in points $T_a$, $T_b$, $T_c$. Show that if the set of points $\left\{A,B,C\right\}$ can be mapped to the set $\left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?

2009 German TST p2 (VAIMO)

Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$

2009 German TST3 p1 (AIMO) (by Gunther Vogel)
Let $ABCD$ be a chordal/cyclic quadrilateral. Consider points $P,Q$ on $AB$ and $R,S$ on $CD$ with $\overline{AP}: \overline{PB} = \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} = \overline{CR}: \overline{RD}.$ How to choose $P,Q,R,S$ such that $\overline{PR} \cdot \overline{AB}+ \overline{QS} \cdot \overline{CD}$ is minimal?

2009 German TST4 p1 (AIMO) (by Christian Reiher)
Let $I$ be the incircle centre of triangle $ABC$ and $\omega$ be a circle within the same triangle with centre $I.$ The perpendicular rays from $I$ on the sides $\overline{BC}, \overline{CA}$ and $\overline{AB}$ meets $\omega$ in $A', B'$ and $C'.$ Show that the three lines $AA', BB'$ and $CC'$ have a common point.

2009 German TST6 p3 (AIMO)
Let $A,B,C,M$ points in the plane and no three of them are on a line. And let $A',B',C'$ points such that $MAC'B, MBA'C$ and $MCB'A$ are parallelograms:
(a) Show that $\overline{MA} + \overline{MB} + \overline{MC} < \overline{AA'} + \overline{BB'} + \overline{CC'}.$
(b) Assume segments $AA', BB'$ and $CC'$ have the same length. Show that $2 \left(\overline{MA} + \overline{MB} + \overline{MC} \right) \le \overline{AA'} + \overline{BB'} + \overline{CC'}.$ When do we have equality?

2010 German TST p1 (VAIMO) 
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

2010 German TST5 p1 (AIMO) 

In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

2011 German TST p1 (VAIMO) 
Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$.  Prove that $CNMF$ is a cyclic quadrilateral.

2012 German TST p2 (VAIMO) 

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the  segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

2013 German TST p3 (VAIMO) 
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

2013 German TST p4 (VAIMO) 
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$

2014 German TST p2 (VAIMO)
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$. Prove $AN \cdot NC = CD \cdot BN.$

2015 German TST p3 (VAIMO) (CGMO 2014)

Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$. Prove $|AP|=|AQ|$.

2015 German TST p5 (VAIMO)

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.

Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.

2016 German TST p1 (VAIMO)

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

2017 German TST p2 (VAIMO)

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

[2018 - 2020 those where SL problems]

source: https://www.mathe-wettbewerbe.de/imo/imo-auswahl-aufgabenarchiv